I'll use Murd's format with a few 'extras' at the end.

Battles: FFXIII-2 is, as Murd mentioned, an enhanced version of FFXIII. It gets the marks for this one by default. Although not having Haste and Hastega, not to mention only a handful of monsters having one of the En-[element] spells... I don't like that at all. I miss buffing, and there is very little to do for a Synergist in FFXIII. So I wouldn't say that FFXIII-2 stormed it, because I loved having those skills so much in the previous game. It's close, but XIII-2 takes the point.

Characters: FFXIII's characters were far more interesting than Serah ever was, for me, and Noel is not good enough to keep up with the number of characters we saw develop in XIII. With that in mind, FFXIII wins this one.

Story: Erk, FFXIII-2 is not strong here at all. FFXIII wasn't generally strong here, either, for many people - but I think it still easily beats FFXIII-2. The (initial) endings for each game were both average at best, but I liked XIII's overall. It felt right, and despite there being a bit of deus ex machina going on, this is something I would say is not new for the Final Fantasy series.

Crystarium: I don't think either is that much better than the other, actually. You had a little more choice involved in XIII, but not much. The system for monsters, though, is pretty fantastic. So it's hard to say. I'll call it a draw.

Sidequests: Massive win for XIII-2. Goes without saying, it's that big a difference.

Gameplay: Since XIII was fight & watch cinematics, there just isn't enough to compete with XIII-2's fight, watch cinematics, race chocobos, take on a general knowledge quiz, explore for fragments, etc. etc. XIII-2 wins this easily.

Final score? XIII with two points, XIII-2 with three. Close, really, and shows that the two games - while similar in some ways - are incredibly different, and the strengths of each game reflect this well. It should really have been 2-2 and a draw over which is the better game if I'd made Battles an even thing over the lack of buffs, but the monsters addition is more valuable than speed.

Oh, and I'll add in that the music is a draw, too. Sound effects go to XIII-2, because I don't have to put up with overly-loud footsteps like I did in XIII. Seriously, some of those footsteps just made you forget there was even music playing. It would be easy to give the music score to XIII-2 because the songs are more memorable, but I think this is almost purely down to the fact that the music is overpowered by the sound effects in XIII.